Ch. 2 Key Equations - University Physics Volume 1 _ OpenStax.pdf

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Unformatted text preview: 10/6/2021 Ch. 2 Key Equations - University Physics Volume 1 | OpenStax Key Equations Multiplication by a scalar (vector equation) ⃗ ⃗ = Multiplication by a scalar (scalar equation for magnitudes) = || Resultant of two vectors ⃗ ⃗ ⃗ = + Commutative law ⃗ ⃗ ⃗ ⃗ + = + Associative law ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ( + ) + = + ( + ) Distributive law ⃗ ⃗ ⃗ 1 + 2 = ( 1 + 2 ) The component form of a vector in two dimensions ⃗ = ˆ + ˆ Scalar components of a vector in two dimensions = − { = − Magnitude of a vector in a plane 2 2 ‾‾‾‾‾‾‾ ‾ = √ + The direction angle of a vector = tan −1 ( ) 1/4 10/6/2021 Ch. 2 Key Equations - University Physics Volume 1 | OpenStax in a plane Scalar components of a vector in a plane Polar coordinates in a plane = cos { = sin = cos { = sin The component form of a vector in three dimensions ⃗ = ˆ + ˆ + ˆ The scalar zcomponent of a vector in three dimensions = − Magnitude of a vector in three dimensions 2 2 2 ‾‾‾‾‾‾‾‾‾‾‾ ‾ = √‾ + + Distributive property ⃗ ⃗ ⃗ ⃗ ( + ) = + Antiparallel vector to ⃗ Equal vectors ⃗ ˆ − = − ˆ − − ˆ ⃗ ⃗ = ⇔ ⎧ = ⎪ ⎨ = ⎪ ⎩ = Components of 2/4 10/6/2021 Ch. 2 Key Equations - University Physics Volume 1 | OpenStax the resultant of N vectors ⎧ ⎪ = 1 + 2 + … + = ∑ ⎪ =1 ⎪ ⎪ ⎨ = ⎪ ∑ = 1 + 2 + … + =1 ⎪ ⎪ ⎪ = ⎩ ∑ = 1 + 2 + … + =1 General unit vector ˆ = Definition of the scalar product ⃗ ⃗ · = cos Commutative property of the scalar product ⃗ ⃗ ⃗ ⃗ · = · Distributive property of the scalar product ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ · ( + ) = · + · Scalar product in terms of scalar components of vectors ⃗ ⃗ · = + + Cosine of the angle between two vectors Dot products of unit vectors ⃗ cos = ⃗ ⃗ · ˆ · ˆ = ˆ · ˆ = ˆ · ˆ = 0 3/4 10/6/2021 Ch. 2 Key Equations - University Physics Volume 1 | OpenStax Magnitude of the vector product (definition) ∣ ⃗ ⃗ ∣ ∣ × ∣ = sin ∣ ∣ Anticommutative property of the vector product ⃗ ⃗ ⃗ ⃗ × = − × Distributive property of the vector product ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ × ( + ) = × + × Cross products of unit vectors ⎧ˆ ˆ × = +ˆ , ⎪ ˆ ⎨ˆ × ˆ = + , ⎪ ˆ ˆ ⎩ˆ × = +. The cross product in terms of scalar components of vectors ⃗ ⃗ × = ( − )ˆ + ( − )ˆ 4/4 ...
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